Another Post on Beauty

This time, however, we’re taking a different tack.

Sometimes, we take beauty for granted. For years, Euler’s formula (specifically, the special case of the formula where x = π, which is known as Euler’s identity)  has been known as the most beautiful theorum in mathematics.

Richard Feynman himself called Euler’s formula “our jewel” and “one of the most remarkable, almost astounding, formulas in all of mathematics.”

What is it, exactly?

Well, Euler’s formula looks a little something like this:

e^{ix} = \cos x + i\sin x \!

Euler’s formula demonstrates the deep relationship between complex exponential and trigonometric functions. Basically, the function shows that e^ix traces the unit circle as x ranges through the real numbers.

It’s elegant and lovely.

Now, if we make x = π, Euler’s formula becomes this very simple and beautiful little identity (because sinπ =0 and cosπ = -1, remember?):

e^{i \pi} + 1 = 0\,\!

What makes Euler’s identity so very lovely is that it manages to connect the most fundamental numbers (e, i, π, 1, 0), the most fundamental operations (addition, multiplication, and exponentiation), the most important relation (=), and nothing else.

It’s perfection in numerical form.

And though you might know all about Euler’s identity already, I want you to do what I like to do every so often. Just pause, look at the equation, think about what it means, and marvel in its elegant simplicity.

That equation, dear galleons, is true beauty.

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